I could see something like that happening if the extinction was predictable i.e a giant meteor or something, but it could be tomorrow through a disease for all we know.
I doubt humans will ever colonize anywhere outside of our solar system. The moon and Mars perhaps in time, but that’s it. This is assuming that science fiction doesn’t become reality.
“The moon and Mars perhaps[....]”
You’ve got it all wrong. You are thinking in the pattern of trying to copy Earth, which cannot be done in this Solar System, and its a bad choice for the same reason staying on Earth is a bad choice. The Earth’s gravitational field is a magnet for inviting destructive impacts, and it also make surface to orbit transport prohibitively expensive in energy and financial costs.
Mars has some similar and some even greater problems. The Martian atmosphere is too thin for atmospheric braking of deorbiting spacecraft. The Martian atmosphere is too thin. Increasing the size of the Martian atmosphere is hampered by the advanced losses of atmosphere due to insufficient gravity and lack of a strong magnetic field. The lack of the magnetic field presents problems on the surface with incident radiation from the Sun and outer space. Then there are a plethora of other problems.
The future of most of humanity is among the asteroids. Mining out multiple galleries and levels of the interior of the larger asteroids yields more arable land and freshwater lakes than are found on the Earth in just one large asteroid. There are many of the large asteroids and dwarf planets out there and within reach, and there are countless thousands more of them of smaller sizes. There is enough water out there to fill a multitude of Earth oceans, and this water can be used to crreate vast lakes and seas INSIDE of these asteroids.
Once humans have learned to turn asteroids into habitats, asteroids can be used to transport human communities throughout the galaxy at sub-light speeds. They can spread new colonies along the paths of their journeys.